Well that is not how we teach - it is called "giving you the answers" and that would defeat the purpose of the extra credit work.
I know we all like walk-throughs - however, there are more effective learning styles.

Instead, please show us your attempts at the questions you have trouble with and we can help you work it out for yourself ;)

On 2/3 - the slopes being equal at all times involves differentiation.
4. just asks to compute the transmission coefficient - do you know how to get the power from the wave-functions? Do you know what the wavefunctions represent?

I know Ai+Ar=At. I figure I might be able to find all those variables in an equation with different coefficients and find the answer in the system of equations. I haven't succeeded in finding that equation though. I took the derivative of each equation with respect to t and had x at zero already. After posting this I will look up how to figure the slope of a multivariable function.

It only asks you for the slope of the net disturbances across the junction ... do you get why the amplitudes and the slopes need to be the same? Have you seen a pulse on a spring travel to a heavier spring?

Watch this:

... in particular, look at the disturbance and slopes of the disturbance at the boundaries - particularly where a light and heavy string is joined.

I dont think i get it yet We're looking at the one point the junction where it is simultaneously the pulse on both strings so of course they have to be the same because they are the same thing. Therefore the derivatives would be the same thing. ???

If I hadn't seen a similar animation before I certainly have now. I've probably seen it in real life before too but don't recall at the moment.

dy/dt shows us the slope of the net disturbance in the direction of time?

dy/dt is the speed the disturbance changes at each point. It's the transverse velocity.
You actually seem to be getting there though ... the derivatives at the junction will get you a system of simultaneous equations.

See how this style of teaching works?
There are actually several ways to approach solving the problem, I could just pick my favorite method and spell it out. This way I get to help you with the way you think about it - filling in gaps in your prior understanding as we go - hopefully you'll end up with a deeper understanding that can be used in other areas as well.

I saw my professor today. He showed me that way using dy/dt the transverse velocity which can also be called the particle velocity right. I tried that at first but made a simple error so I kept on trying to figure a different way.

I made another post. I have the first test tomorrow and am figuring I should actually focus on his past tests right now but may like to come back to this thread later.

I am all for having a deeper understanding of reality. Thanks for the help. Please check out my other thread. Lol